To begin at the beginning, assume that we fix non-negative
integers \(g\)  and \(n\)  with \(2g-2+n\ge 1\) . A smooth projective geometrically irreducible
curve \(C\)  of genus \(g\) 
together with \(n\)  distinct labeled
points \(P_1,\ldots ,P_n\)  is called an \(n\) -pointed
curve of genus \(g\) . It is called stable if the group of automorphisms of \(C\) 
fixing \(P_1,\ldots ,P_n\)  is finite.
Here a morphism of \((C,P_1,\ldots ,P_n)\)  to \((D,Q_1,\ldots ,Q_n)\) 
is a morphism \(\varphi : C \rightarrow D\)  with \(\varphi (P_i)=Q_i\)  for \(i=1,\ldots ,n\) .
The most basic fact here, due to Mumford,
is the existence for \(2g-2+n\ge 1\) 
of a moduli space \({\mathcal {M}}_{g,n}\)  of stable \(n\) -pointed
curves of genus \(g\) , [1]}.
It is an irreducible Deligne-Mumford stack of
dimension \(3g-3+n\)  defined over \(\mathbb {Z}\) . That we have to deal with stacks and not just
with varieties, is due to the fact that curves can have non-trivial automorphism groups.
